Matching regular expressions including word boundary symbols

ABSTRACT

According to an example, a method for matching regular expressions including word boundary symbols includes receiving an input string and receiving a regular expression including a word boundary symbol. The method further includes transforming, by a processor, the regular expression into an automaton such that a set of strings accepted by the automaton is the same as a set of strings described by the regular expression. The method also includes processing the input string by the automaton to determine if the input string matches the regular expression.

BACKGROUND

Regular expressions can be used to provide a concise and formal way ofdescribing a set of strings over an alphabet. A regular expressionmatches a string if the string belongs to the set described by theregular expression. Regular expression matching may be used, forexample, by command shells, programming languages, text editors, andsearch engines to search for text within a document. Regular expressionscan include word boundary symbols that match boundaries between word andnon-word characters. The worst-case matching times of known techniquesfor checking whether an input string matches a regular expression withword boundary symbols can be exponential in the length of the inputstring.

BRIEF DESCRIPTION OF DRAWINGS

Features of the present disclosure are illustrated by way of example andnot limited in the following figure(s), in which like numerals indicatelike elements, in which:

FIG. 1 illustrates an architecture of an apparatus for matching regularexpressions including word boundary symbols, according to an example ofthe present disclosure;

FIG. 2 illustrates further details of an automata generation module forthe apparatus for matching regular expressions including word boundarysymbols, according to an example of the present disclosure;

FIG. 3 illustrates details of an automaton M_(B) that accepts strings ina language defined in an example of the present disclosure;

FIG. 4 illustrates details of an automaton M₁ that accepts strings ofsymbols in an alphabet which is the alphabet over which a regularexpression E is defined, with the addition of word and non-word boundarysymbols \b and \B, respectively, according to an example of the presentdisclosure;

FIG. 5 illustrates details of an automaton M₂ that accepts the samestrings as the automaton M₁{circle around (x)}M_(B), where the automatonM₁ is as shown in FIG. 4, according to an example of the presentdisclosure;

FIG. 6 illustrates details of an automaton M₃ that accepts the samestrings as the automaton obtained by replacing each edge in theautomaton M₂ of FIG. 5 labeled by the word and non-word boundary symbols\b and \B, respectively, by an ε edge, according to an example of thepresent disclosure;

FIG. 7 illustrates a method for matching regular expressions includingword boundary symbols, according to an example of the presentdisclosure;

FIG. 8 illustrates further details of the method for matching regularexpressions including word boundary symbols, according to an example ofthe present disclosure; and

FIG. 9 illustrates a computer system, according to an example of thepresent disclosure.

DETAILED DESCRIPTION

For simplicity and illustrative purposes, the present disclosure isdescribed by referring mainly to examples. In the following description,numerous specific details are set forth in order to provide a thoroughunderstanding of the present disclosure. It will be readily apparenthowever, that the present disclosure may be practiced without limitationto these specific details. In other instances, some methods andstructures have not been described in detail so as not to unnecessarilyobscure the present disclosure.

Throughout the present disclosure, the terms “a” and “an” are intendedto denote at least one of a particular element. As used herein, the term“includes” means includes but not limited to, the term “including” meansincluding but not limited to. The term “based on” means based at leastin part on.

Regular expressions are a formal way to describe a set of strings overan alphabet. Regular expression matching is the process of determiningwhether a given string (for example, a string of text in a document)matches a given regular expression. That is, whether the given string isin the set of strings that the regular expression describes. Regularexpressions can contain word and non-word symbols that match any wordcharacter or any non-word character, respectively, and also can containword boundary symbols that match the boundaries between word andnon-word characters. The symbol \w matches word characters, and thesymbol \W matches non-word characters. The word characters are the upperand lower case letters, numbers and the underscore character ‘_’.

The word boundary symbol \b is used in a regular expression to specifythat a word boundary occurs in a particular place in strings matchingthe regular expression. Similarly, the non-word boundary symbol \B isused in a regular expression to specify a place where a non-wordboundary occurs. Word boundaries occur at the following three positionsin a string. First, a word boundary occurs before the first character ina string if the first character is a word character. Second, a wordboundary occurs after the last character in a string if the lastcharacter is a word character. Third, a word boundary occurs between twocharacters in a string, where one character is a word character and theother character is a non-word character. Non-word boundaries occur atpositions in a string that are not word boundaries.

For example, consider the regular expression “.*ice\b.*”. Thesubexpression “.*” matches any substring. Therefore, the stringsmatching this regular expression are those that contain a substring icefollowed immediately by a word boundary. Thus, this regular expressionmatches the strings twice and ice cream, but not the string ice9.

The word boundary symbols may also represent expressions that generallyrepresent different types of alphabets, characters or other objects. Forexample, the symbols \w(S), \W(S), \b(S) and \B(S) may be used for anysubset S of the alphabet over which the input string is defined, where\w(S) matches the characters in S, \W(S) matches the characters not inS, \b(S) indicates a boundary between a character in S and a characternot in S, and \B(S) indicates a boundary that is not matched by \b(S).Further, \b(S) matches the beginning of a string if and only if (i.e.,iff) the first character of the string is not in S, and the end of thestring iff the last character of the string is not in S. These symbolshave the same behavior as \w, \W, \b, and \B when the set S is the setof word characters. For example, the symbols \w(S) and \W(S) mayrepresent a partition into numbers and non-numbers. In such a case, thesymbols \b(S) and \B(S) may represent boundaries for the numbers andnon-numbers, respectively.

According to an example, an apparatus and method are described formatching regular expressions including word boundary symbols. Theapparatus for matching regular expressions including word boundarysymbols and method provide for implementation of regular expressionsthat include word boundary symbols in automata. Generally, a method formatching regular expressions including word boundary symbols includesreceiving an input string and receiving a regular expression including aword boundary symbol. The method further includes transforming, by aprocessor, the regular expression into an automaton such that a set ofstrings accepted by the automaton is the same as a set of stringsdescribed by the regular expression. The method also includes processingthe input string by the automaton to determine if the input stringmatches the regular expression.

According to an example, an apparatus for matching regular expressionswith word boundary symbols includes a memory storing a module comprisingmachine readable instructions to receive an input string, and receive aregular expression including a word boundary symbol. The module furthercomprises machine readable instructions to transform the regularexpression into an automaton such that a set of strings accepted by theautomaton is the same as a set of strings described by the regularexpression. The apparatus further comprises a processor to implement themodule.

According to another example, a non-transitory computer readable mediumhaving stored thereon machine readable instructions for matching regularexpressions including boundary symbols is also described. The machinereadable instructions when executed cause a computer system to receivean input string, receive a regular expression including a boundarysymbol, and transform, by a processor, the regular expression into anautomaton such that a set of strings accepted by the automaton is thesame as a set of strings described by the regular expression.

The apparatus for matching regular expressions including word boundarysymbols and method provide a scalable technique for matching regularexpressions including word boundary symbols.

FIG. 1 illustrates an architecture of an apparatus for matching regularexpressions including word boundary symbols 100, according to anexample. Referring to FIG. 1, the apparatus 100 is depicted as includingan input module 101 to receive a regular expression including a wordboundary symbol. The regular expression is transformed to an automatonby an automata generation module 102. Referring to FIGS. 1 and 2, asdescribed in further detail below, the automata generation module 102generates the automata M₁, M₂, and M₃, with the regular expression beingtransformed to the final automaton M₃ such that the set of stringsaccepted by the automaton M₃ is equal to the set of strings specified bythe regular expression. For example, the automata generation module 102includes the M₁ automaton generation module 103, the M₂ automatongeneration module 104, and the M₃ automaton generation module 105 torespectively generate the automata M₁, M₂, and M₃. A comparison module106 is to receive input strings, and determine whether the input stringsmatch the regular expression. If an input string does not match theregular expression, the output module 107 indicates accordingly.However, if an input string matches the regular expression, an outputmodule 108 indicates the match. In this manner, the regular expressionmay be transformed to an automaton by the automata generation module102, and then matched to many different input strings by the comparisonmodule 106.

The modules 101-108, and other components of the apparatus 100 thatperform various other functions in the apparatus 100, may comprisemachine readable instructions stored on a computer readable medium. Inaddition, or alternatively, the modules 101-108, and other components ofthe apparatus 100 may comprise hardware or a combination of machinereadable instructions and hardware.

In order to match a regular expression E containing a word boundarysymbol, the regular expression E is transformed to an automaton by theautomata generation module 102. Referring to FIGS. 1 and 2, as describedin further detail below, the automata generation module 102 generatesthe automata M₁, M₂, and M₃, with the regular expression E beingtransformed to the final automaton M₃ such that the set of stringsaccepted by M₃ is the same as the set of strings specified by theregular expression E.

For the automata generation module 102, an automaton M may be defined asa five-tuple M=(Q, Σ, Δ, I, F), where Q is a finite set of states, Σ isa finite alphabet, Δ: Q×Σ→Q maps a current state and input pair to asubsequent state, I is a set of initial states, and F is a set of final(or accepting) states. An element of Δ is denoted by a triple (p, a, q),where p ε Q is the current state, a ε Σ is an input character, and q ε Qis the subsequent state.

For a regular expression E including a word boundary symbol, in order totransform the regular expression E into the final automaton M₃, the M₁automaton generation module 103 generates an automaton M₁. The automatonM₁ accepts strings of symbols in an alphabet, which is the alphabet overwhich the regular expression E is defined, with the addition of the wordand non-word boundary symbols \b and \B, respectively. The alphabetincluding the word and non-word boundary symbols \b and \B is designatedan extended alphabet. The automaton M₁ accepts a string of symbols inthis extended alphabet if and only if the string is in the set describedby the regular expression over this extended alphabet that issyntactically identical to the regular expression E (which excludes theword and non-word boundary symbols \b and \B), but where the symbols \band \B are treated as ordinary alphabet symbols rather than beinginterpreted as word or non-word boundary requirements. The automaton M₁is used by the M₂ automaton generation module 104 to generate theautomaton M₂ that is defined as the cross-product of automata M₁ andM_(B) (i.e., M₂=M₁{circle around (x)}M_(B)). The automaton M_(B) isdefined as an automaton that accepts all strings over the extendedalphabet except those that contain substrings of the form \B\b, \b\B,\w\b\w, \w\B\W, \W\b\W, or \W\B\w, those that begin with \b\W or \B\w,and those that end in \W\b or \w\B. The designation L_(B) may be usedfor the set of strings accepted by the automaton M_(B). The M₃ automatongeneration module 105 generates the automaton M₃ by replacing each edgein the automaton M₂ labeled by the boundary symbols \b or \B by an εedge. An ε edge is an edge which can be taken without consuming acharacter in an input string. The ε edges can be eliminated by firstcomputing for each state y, the set of states e(y) that can be reachedfrom state y by traversing only ε edges. The ε edges are then deleted.For each edge from state x to state y labeled with an input character(e.g., ‘a’), new edges are added from state x to every state in e(y)labeled with input character ‘a’. The automaton M₃ is used by thecomparison module 106 to determine whether an input string is in the setof strings specified by the regular expression E.

As discussed above, the automaton M₁ is used by the M₂ automatongeneration module 104 to generate the automaton M₂ that is defined asthe cross-product of automata M₁ and M_(B) (i.e., M₂=M₁{circle around(x)}M_(B)). Generally, for an automaton M defined as a five-tuple M=(Q,Σ, Δ, I, F), the cross product of the automata M₁ and M_(B) is definedas follows. For example, let the automaton M₁=(Q₁, Σ, Δ₁, I₁, F₁) be theautomaton corresponding to the language L₁

Σ*, and the automaton M_(B)=(Q_(B), Σ, Δ_(B), I_(B), F_(B)) be theautomaton corresponding to the language L_(B)

Σ*. The cross-product of the automata M₁ and M_(B) is:M ₁ {circle around (x)}M _(B)=(Q ₁ ×Q _(B) ,Σ,Δ,I ₁ ×I _(B) ,F ₁ ×F_(B))  Equation (1)For Equation (1), ((p₁, p_(B)), a, (q₁, q_(B))) ε Δ if and only if (p₁,a, q₁) ε Δ₁ and (p_(B), a, q_(B)) ε Δ_(B). The set of strings acceptedby the automaton M₁{circle around (x)}M_(B) is equal to L₁ ∩ L_(B).

In order to prove that the set of strings accepted by the automaton M₃is the same as the set specified by the regular expression E, consider Lto denote the set of strings specified by the regular expression E. Theset of strings accepted by the automaton M₁ is L₁=L⁺ ∪ X, where L⁺ isthe subset of L₁ such that if each instance of \b and \B in the stringis deleted, the resulting string is in L. It follows thatL⁺|_(\b=ε, \B)=_(ε) is equal to L. The set of strings X includes thestrings in L₁ that do not obey the boundary rules relative to \b and \B.For example, suppose that E=.\b., and the alphabet Σ over which E isdefined contains a, b and =. By definition, the metacharacter “.”matches any character in Σ, and it follows that for example L⁺ containsthe string a\b=, while X contains the string a\bb. By definition, themetacharacter “.” matches any character in Σ. The set of strings L₂ thatare accepted by the automaton M₂, which is defined as the cross-productof the automata M₁ and M_(B) (i.e., M₂=M₁{circle around (x)}M_(B)),satisfies L₂=L₁ ∩ L_(B)=L⁺. The automaton M₃ is produced by replacingeach edge in the automaton M₂ labeled by the boundary symbols \b or \Bby an E edge (which can then be eliminated as discussed above), andtherefore the set of strings accepted by the automaton M₃ isL₂|_(\b=ε, \B)=_(ε)=L. Therefore the set of strings accepted by theautomaton M₃ is equal to the set of strings specified by the regularexpression E.

As discussed above, the automaton M_(B) accepts the strings in thelanguage L_(B). For example, it can be seen that the string “\bThe dog\bcha\Bsed the cat.\B” is in the language L_(B). It can also be seen thatthe word and non-word boundary symbols \b and \B follow the requirementsfor word and non-word boundaries. For example, it can be seen that inthe substring “dog\b cha\Bsed”, the word boundary symbol \b occursbetween a word character (“g”) and a non-word character (a space) in thestring, and the non-word boundary symbol \B occurs between two wordcharacters in the string.

The automaton M_(B) that accepts strings in the language L_(B) isgraphically shown at 120 in FIG. 3. For FIG. 3, an initial state for theautomaton M_(B) is shown at 121, a transition state is shown at 122 anda final state is shown at 123. For a string input into the automatonM_(B) at the initial state 121, once the complete string is processedthrough the automaton M_(B), if the last state is a final state such asfinal state 123, then the string is considered to match (i.e., beaccepted by) the automaton M_(B) and the regular expression representedby the automaton M_(B) (i.e., the string is in the language L_(B)).Similarly, once the complete string is processed through the automatonM_(B), if the last state is a transition state such as transition state122, then the string does not match the automaton M_(B) and the regularexpression represented by the automaton M_(B) (i.e., the string is notin the language L_(B)). Likewise, if the complete string cannot beprocessed by the automaton M_(B), then the string does not match theautomaton M_(B) and the regular expression represented by the automatonM_(B) (i.e., the string is not in the language L_(B)). For the automatonM_(B), the first character of an input string input at the initial state121 has the possibilities of being a non-word boundary symbol \B at 124,a word boundary symbol \b at 125, a word character represented by \w at126 or a non-word character represented by \W at 127. Subsequentcharacters in the input string can be processed in a similar manner todetermine if the input string is in the language L_(B).

Referring to FIGS. 1-7, an example of an application of the apparatus100 for matching regular expressions including word boundary symbols isdescribed for the regular expression (\w| . . . \b)*. For the regularexpression (\w| . . . \b)*, the * indicates to perform the operation inthe parenthesis zero or more times. The | indicates to either performthe operation on the left or right hand side. The \w indicates matchingof a word character. The period “.” indicates matching any character.The word boundary symbol \b indicates a word boundary.

For the example of the regular expression (\w| . . . \b)*, FIG. 4illustrates the automaton M₁ at 130 generated by the M₁ automatongeneration module 103. For FIG. 4, any non-accepting states whoseoutgoing transitions are all self-loops or any edges that are incidentto non-accepting states are omitted. For the regular expression (\w| . .. \b)*, starting at state-0 at 131 of FIG. 4, an empty string can matchthe regular expression (\w| . . . \b)* and therefore make the automatonremain at state-0. In order to move the automaton to state-3 at 132, aninput string can include a non-word character that matches the firstperiod “.” in the regular expression (\w| . . . \b)*. After state-3, theinput string can include a word character or a non-word charactermatching the second period “.” in the regular expression to move theautomaton to state-5 at 133. After state-5, a boundary symbol \b in theinput string returns the automaton to state-0, a final state.Alternatively, in order to move to state-1 at 134, the input string caninclude a word character that matches the first period “.” of theregular expression or the word symbol \w on the left of the |. Afterstate-1, the input string can include a word character matching thesecond period “.” to move the automaton to state-2 at 135. Afterstate-2, a boundary symbol \b returns the automaton to state-0, a finalstate.

Referring to FIG. 5, the automaton M₁ is used by the M₂ automatongeneration module 104 to generate the automaton M₂ at 140 that isdefined as the cross-product of automata M₁ and M_(B) (i.e.,M₂=M₁{circle around (x)}M_(B)). For FIG. 5, any non-accepting stateswhose outgoing transitions are all self-loops or any edges that areincident to the non-accepting states are omitted. Further, it is notedthat the state numbering of FIG. 5 does not correspond to the statenumbering of FIG. 4. Referring to FIG. 4, assuming a string includes theinitial sequence ==\b= that moves the automaton in FIG. 4 to the states0, 3, 5, 0, and 3, the same string will move the automaton in FIG. 5 tostates 0, 3, 5, and 28, respectively at 141, 142, 143, and 144. However,at state-28 of FIG. 5, since the automaton M₂ accounts for boundaryconditions (i.e., M₂=M₁{circle around (x)}M_(B)), and one of theboundary conditions excludes strings that contain subsequences of theform \W\b\W, the string including the sequence ==\b= does not match theautomaton M₂, and in fact from state-28 in FIG. 5 there is no outgoingarrow with a label matching the character = which is next in the string.

Referring to FIG. 6, the M₃ automaton generation module 105 generatesthe automaton M₃ at 150 by replacing each edge in the automaton M₂labeled by the boundary symbols \b or \B by an e edge. The resultingautomaton can be minimized. The automaton M₃ is used by the comparisonmodule 106 to determine whether an input string is in the set of stringsspecified by the regular expression E. For FIG. 6, any non-acceptingstates whose outgoing transitions are all self-loops or any edges thatare incident to the non-accepting states are omitted. For example, withregard to replacement of the boundary symbols \b and \B by an E edge,state-4 at 151 of FIG. 6 corresponds to states-5, 28, 34 and 41 of FIG.5. For example, when the boundary symbols \b and \B are replaced by εedges, the initial state (i.e., state-0) in FIG. 6 corresponds to thestates (i.e., states-0, 34 and 41) reachable by ε moves in FIG. 5. Theset of states reachable from the set {0, 34, 41} of FIG. 5 on a \W isthe set {3, 6}. Further including the states reachable by E moves, theset of states in FIG. 5 is {3, 6, 34, 41}. Therefore state-2 of FIG. 6corresponds to the set of states {3, 6, 34, 41} of FIG. 5. In otherwords, the set of states reachable from the set {3, 6, 34, 41} of FIG. 5on a \W is the set {5, 6}. Further, including the states reachable fromstates {5, 6} by ε moves, the set of states in FIG. 5 that can bereached from state-0 with two 1W moves and zero or more ε moves in someorder is {5, 6, 28, 34, 41}. Therefore, state-4 of FIG. 6 corresponds tothe set of states {5, 6, 28, 34, 41} of FIG. 5. For a string input intothe automaton M₃ at initial state-0 at 152, once the complete string hasbeen processed by the automaton M₃, if the last state is a final statesuch as final states-1 or 3, respectively at 153 and 154, then thestring is considered to match the automaton M₃ and the regularexpression represented by the automaton M₃. Similarly, once the completestring is processed by the automaton M₃, if the last state is atransition state such as transition states 2, 4 or 6, respectively at155, 151 and 156, then the string does not match the automaton M₃ andthe regular expression represented by the automaton M₃. Likewise, if thecomplete string cannot be processed by the automaton M₃, then the stringdoes not match the automaton M₃ and the regular expression representedby the automaton M₃. For example, the string =a matches the regularexpression represented by the automaton in FIG. 6 by proceeding fromstate-0 to state-2 to final state-3, because = is a non-word characterand a is a word character.

FIGS. 7 and 8 illustrate flowcharts of methods 200 and 300 for matchingregular expressions including word boundary symbols, corresponding tothe example of the apparatus 100 for matching regular expressionsincluding word boundary symbols whose construction is described indetail above. The methods 200 and 300 may be implemented on theapparatus 100 for matching regular expressions including word boundarysymbols with reference to FIGS. 1-6 by way of example and notlimitation. The methods 200 and 300 may be practiced in other apparatus.

Referring to FIG. 7, for the method 200, at block 201, a regularexpression including a word boundary symbol is received. For example,referring to FIG. 1, the input module 101 receives a regular expressionincluding a word boundary symbol.

At block 202, the regular expression is transformed into an automatonsuch that a set of strings accepted by the automaton is the same as aset of strings described by the regular expression. For example,referring to FIG. 1, the regular expression is transformed to anautomaton by the automata generation module 102.

At block 203, an input string is received. For example, referring toFIG. 1, the comparison module 106 receives an input string.

At block 204, the input string is processed by the automaton todetermine if the input string matches the regular expression. Forexample, referring to FIG. 1, the comparison module 106 determineswhether the input string matches the regular expression.

At block 205, an output indicating if the input string matches or doesnot match the regular expression is generated. For example, referring toFIG. 1, if the input string matches the regular expression, the outputmodule 108 indicates the match. If the input string does not match theregular expression, the output module 107 indicates accordingly.

Referring to FIG. 8, for the method 300, at block 301, a regularexpression including a word boundary symbol and an input string arereceived.

At block 302, the regular expression is transformed into a firstautomaton M₁ (i.e., a first intermediate automaton) which acceptsstrings of symbols in an alphabet which is the alphabet over which theregular expression is defined, with the addition of the symbols \b and\B. The first automaton M₁ accepts a string in this extended alphabet ifand only if it is in the set described by the regular expression overthis extended alphabet that is syntactically identical to E (whichexcludes the word and non-word boundary symbols \b and \B), but wherethe symbols \b and \B are treated as ordinary alphabet symbols ratherthan being interpreted as word boundary symbols. For example, referringto FIGS. 1 and 2, the M₁ automaton generation module 103 generates theautomaton M₁.

At block 303, the first automaton M₁ is used to generate a secondautomaton M₂ (i.e., a second intermediate automaton) determined by across-product of the first automaton and an automaton M_(B) that acceptsall strings over the extended alphabet except those that includesubstrings of the form \B\b, \b\B, \w\b\w, \w\B\W, \W\b\W, or \W\B\w,strings that begin with \b\W or \B\w, or strings that end in \W\b or\w\B, where \b is a word boundary symbol, \B is a non-word boundarysymbol, \w represents any word character and \W represents any non-wordcharacter. The first automaton is used to generate a second automatonM₂, which is the cross-product of automata M₁ and M_(B) (i.e.,M₂=M₁{circle around (x)}M_(B)).

At block 304, a third automaton M₃ (i.e., a final automaton) isgenerated by replacing each edge in the second automaton M₂ labeled bythe word or non-word boundary symbols \b or \B, respectively, by an εedge. For example, referring to FIGS. 1 and 2, the M₃ automatongeneration module 105 generates the automaton M₃ by replacing each edgein the automaton M₂ labeled by the boundary symbols \b and \B by an εedge.

At block 305, the input string is processed by the third automaton M₃and if the input string is accepted by the third automaton M₃, then itmatches the regular expression. For example, referring to FIG. 1, theautomaton M₃ is used by the comparison module 106 to determine whetherthe input string is in the set of strings specified by the regularexpression.

FIG. 9 shows a computer system that may be used with the examplesdescribed herein. The computer system represents a generic platform thatincludes components that may be in a server or another computer system.The computer system may be used as a platform for the apparatus 100. Thecomputer system may execute, by a processor or other hardware processingcircuit, the methods, functions and other processes described herein.These methods, functions and other processes may be embodied as machinereadable instructions stored on a computer readable medium, which may benon-transitory, such as hardware storage devices (e.g., RAM (randomaccess memory), ROM (read only memory), EPROM (erasable, programmableROM), EEPROM (electrically erasable, programmable ROM), hard drives, andflash memory).

The computer system includes a processor 402 that may implement orexecute machine readable instructions performing some or all of themethods, functions and other processes described herein. Commands anddata from the processor 402 are communicated over a communication bus404. The computer system also includes a main memory 406, such as arandom access memory (RAM), where the machine readable instructions anddata for the processor 402 may reside during runtime, and a secondarydata storage 408, which may be non-volatile and stores machine readableinstructions and data. The memory and data storage are examples ofcomputer readable mediums. The memory 406 may include modules 420including machine readable instructions residing in the memory 406during runtime and executed by the processor 402. The modules 420 mayinclude the modules 101-108 of the apparatus shown in FIG. 1.

The computer system may include an I/O device 410, such as a keyboard, amouse, a display, etc. The computer system may include a networkinterface 412 for connecting to a network. Other known electroniccomponents may be added or substituted in the computer system.

What has been described and illustrated herein is an example along withsome of its variations. The terms, descriptions and figures used hereinare set forth by way of illustration only and are not meant aslimitations. Many variations are possible within the spirit and scope ofthe subject matter, which is intended to be defined by the followingclaims—and their equivalents—in which all terms are meant in theirbroadest reasonable sense unless otherwise indicated.

What is claimed is:
 1. A method for matching regular expressionsincluding word boundary symbols, the method comprising: receiving aninput string; receiving a regular expression including a word ornon-word boundary symbol; transforming, by a processor, the regularexpression into an automaton such that a set of strings accepted by theautomaton is the same as a set of strings described by the regularexpression, wherein the automaton is a final automaton derived from aplurality of intermediate automata, and wherein transforming the regularexpression into the final automaton further includes transforming theregular expression into a first intermediate automaton that accepts astring over an extended alphabet including word and non-word boundarysymbols; and processing the input string by the automaton to determineif the input string matches the regular expression.
 2. The method ofclaim 1, wherein transforming the regular expression into a firstintermediate automaton that accepts a string over an extended alphabetincluding word and non-word boundary symbols further comprisestransforming the regular expression into the first intermediateautomaton that accepts the string over the extended alphabet includingword and non-word boundary symbols if and only if: the string over theextended alphabet is in a set of strings described by the regularexpression over the extended alphabet, and the regular expression overthe extended alphabet is syntactically identical to the regularexpression excluding the word and non-word boundary symbols, but wherethe word and non-word boundary symbols are treated as ordinary alphabetsymbols as opposed to being interpreted as the word and non-wordboundary symbols.
 3. The method of claim 2, wherein transforming theregular expression into the final automaton further comprises: using thefirst intermediate automaton to generate a second intermediate automatondetermined by a cross-product of the first intermediate automaton and anautomaton that accepts strings over the extended alphabet but does notaccept strings over the extended alphabet that include substrings of theform selected from \B\b, \b\B, \w\b\w, \w\B\W, \W\b\W, and \ W\B\w,strings that begin with \b\W or \B\w, or strings that end in \ W\b or\w\B, where \b is a word boundary symbol, \B is a non-word boundarysymbol, \w represents any word character and \W represents any non-wordcharacter.
 4. The method of claim 3, wherein transforming the regularexpression into the final automaton further comprises: generating thefinal automaton by replacing an edge in the second intermediateautomaton based on the word or non-word boundary symbols \b or \B,respectively, by an ε edge, where the ε edge is an edge which can betaken without consuming a character in the input string.
 5. The methodof claim 4, further comprising: deleting ε edges in the final automaton.6. An apparatus for matching regular expressions including word boundarysymbols comprising: a memory storing a module comprising machinereadable instructions to: receive an input string; receive a regularexpression including a word or non-word boundary symbol; and transformthe regular expression into an automaton such that a set of stringsaccepted by the automaton is the same as a set of strings described bythe regular expression, wherein the automaton is a final automatonderived from a plurality of intermediate automata, and whereintransforming the regular expression into the final automaton furtherincludes transforming the regular expression into a first intermediateautomaton that accepts a string over an extended alphabet including wordand non-word boundary symbols; and a processor to implement the module.7. The apparatus of claim 6, further comprising machine readableinstructions to: process the input string by the automaton to determineif the input string matches the regular expression.
 8. The apparatus ofclaim 6, wherein transforming the regular expression into a firstintermediate automaton that accepts a string over an extended alphabetincluding word and non-word boundary symbols further comprises machinereadable instructions to transform the regular expression into the firstintermediate automaton that accepts the string over the extendedalphabet including word and non-word boundary symbols if and only if:the string over the extended alphabet is in a set of strings describedby the regular expression over the extended alphabet, and the regularexpression over the extended alphabet is syntactically identical to theregular expression excluding the word and non-word boundary symbols, butwhere the word and non-word boundary symbols are treated as ordinaryalphabet symbols as opposed to being interpreted as the word andnon-word boundary symbols.
 9. The apparatus of claim 8, whereintransforming the regular expression into the final automaton furthercomprises machine readable instructions to: use the first intermediateautomaton to generate a second intermediate automaton determined by across-product of the first intermediate automaton and an automaton thataccepts strings over the extended alphabet but does not accept stringsover the extended alphabet that include substrings of the form selectedfrom \B\b, \b\B, \w\b\w, \w\B\W, \W\b\W, and \ W\B\w, strings that beginwith \b\W or \B\w, and strings that end in \ W\b or \w\B, where \b is aword boundary symbol, \B is a non-word boundary symbol, \w representsany word character and \W represents any non-word character.
 10. Theapparatus of claim 9, wherein transforming the regular expression intothe final automaton further comprises machine readable instructions to:generate the final automaton by replacing an edge in the secondintermediate automaton based on the word or non-word boundary symbols \bor \B, respectively, by an ε edge, where the ε edge is an edge which canbe taken without consuming a character in the input string.
 11. Anon-transitory computer readable medium having stored thereon machinereadable instructions for matching regular expressions includingboundary symbols, the machine readable instructions when executed causea computer system to: receive an input string; receive a regularexpression including a boundary symbol; and transform, by a processor,the regular expression into an automaton such that a set of stringsaccepted by the automaton is the same as a set of strings described bythe regular expression, wherein the automaton is a final automatonderived from a plurality of intermediate automata, and whereintransforming the regular expression into the final automaton furtherincludes transforming the regular expression into a first intermediateautomaton that accepts a string over an extended alphabet includingboundary symbols.
 12. The non-transitory computer readable medium ofclaim 11, further comprising machine readable instructions to: processthe input string by the automaton to determine if the input stringmatches the regular expression.
 13. The non-transitory computer readablemedium of claim 11, wherein transforming the regular expression into afirst intermediate automaton that accepts a string over an extendedalphabet including boundary symbols further comprises machine readableinstructions to transform the regular expression into the firstintermediate automaton that accepts the string over the extendedalphabet including boundary symbols if and only if: the string over theextended alphabet is in a set of strings described by the regularexpression over the extended alphabet, and the regular expression overthe extended alphabet is syntactically identical to the regularexpression excluding the boundary symbol, but where the boundary symbolis treated as an ordinary alphabet symbol as opposed to beinginterpreted as the boundary symbol.
 14. The non-transitory computerreadable medium of claim 13, wherein transforming the regular expressioninto the final automaton further comprises machine readable instructionsto: use the first intermediate automaton to generate a secondintermediate automaton determined by a cross-product of the firstintermediate automaton and an automaton that accepts strings over theextended alphabet but does not accept strings over the extended alphabetthat include substrings of the form selected from \B\b, \b\B, \w\b\w,\w\B\W, \W\b\W, and \ W\B\w, strings that begin with \b\W or \B\w, orstrings that end in \ W\b or \w\B, where \b is a first boundary symbol,\B is a second boundary symbol, \w represents a first set of characterscorresponding to the first boundary symbol and \W represents a secondset of characters corresponding to the second boundary symbol.
 15. Thenon-transitory computer readable medium of claim 14, whereintransforming the regular expression into the final automaton furthercomprises machine readable instructions to: generate the final automatonby replacing an edge in the second intermediate automaton based on theword or non-word boundary symbols \b or \B, respectively, by an ε edge,where the ε edge is an edge which can be taken without consuming acharacter in the input string.